In functional analysis, the Dixmier–Ng theorem is a characterization of when a normed space is in fact a dual Banach space. It was proven by Kung-fu Ng, who called it a variant of a theorem proven earlier by Jacques Dixmier.[1][2]

Dixmier-Ng theorem.[1] Let be a normed space. The following are equivalent:
  1. There exists a Hausdorff locally convex topology on so that the closed unit ball, , of is -compact.
  2. There exists a Banach space so that is isometrically isomorphic to the dual of .

That 2. implies 1. is an application of the Banach–Alaoglu theorem, setting to the Weak-* topology. That 1. implies 2. is an application of the Bipolar theorem.

Applications

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Let   be a pointed metric space with distinguished point denoted  . The Dixmier-Ng Theorem is applied to show that the Lipschitz space   of all real-valued Lipschitz functions from   to   that vanish at   (endowed with the Lipschitz constant as norm) is a dual Banach space.[3]

References

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  1. ^ a b Ng, Kung-fu (December 1971), "On a theorem of Dixmier", Mathematica Scandinavica, 29: 279–280, doi:10.7146/math.scand.a-11054
  2. ^ Dixmier, J. (December 1948), "Sur un théorème de Banach", Duke Mathematical Journal, 15 (4): 1057–1071, doi:10.1215/s0012-7094-48-01595-6
  3. ^ Godefroy, G.; Kalton, N. J. (2003), "Lipschitz-free Banach spaces", Studia Mathematica, 159 (1): 121–141, doi:10.4064/sm159-1-6